Uniqueness of local, analytic solutions to singular ODEs

A system of four ODEs: the singularity analysis

The singularity analysis is carried out for a system of four first-order quadratic ODEs with a parameter, which was proposed recently by Golubchik and Sokolov. A transformation of dependent variables is revealed by the analysis, after which the transformed system possesses the Painlevé property and does not contain the parameter.

Uniqueness of the measure of maximal entropy for singular hyperbolic flows in dimension 3 and more results on equilibrium states

We prove that any 3-dimensional singular hyperbolic attractor admits for any Hölder continuous potential $V$ at most one equilibrium state for $V$ among regular measures. We give a condition on $V$ which ensures that no singularity can be an equilibrium state. Thus, for these $V$'s, there exists a unique equilibrium state and it is a regular measure. Applying this for $V\equiv 0$, we show that any 3-dimensional singular hyperbolic attractor admits a unique measure of maximal entropy.

Quasi-periodic solutions for differential equations with an elliptic-type degenerate equilibrium point under small perturbations

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity.

Particle dynamics with elastic collision at the boundary: existence and partial uniqueness of solutions

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $Ω$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of…

奇异扰动非局部神经场中快行脉冲的存在与唯一性：完整证明

Existence and Uniqueness of Fast Traveling Pulses in Singularly Perturbed Nonlocal Neural Fields With Heaviside Nonlinearities: a Complete Proof

本文通过严格证明，解决了奇异扰动非局部神经场中快行脉冲解的存在性和唯一性问题，提供了新的近似方法和收敛性分析。

Pathwise uniqueness for singular SDEs driven by stable processes

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $α$-stable Lévy processes with values in $\R^d$ having a bounded and $β$-Hölder continuous drift term. We assume $β> 1 - \fracα{2} $ and $α\in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

Dynamics of quasiconformal fields

A uniqueness theorem is established for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every non-critical point of $f$ there passes a unique integral curve.